.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/communications/information/plot_04_shannon_hartley_capacity.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_communications_information_plot_04_shannon_hartley_capacity.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_communications_information_plot_04_shannon_hartley_capacity.py:


The Shannon-Hartley capacity of a noisy band-limited channel
============================================================

In "Communication in the presence of noise" (1949) Shannon proved that a
channel of bandwidth :math:`B` with white Gaussian noise and signal-to-noise
ratio :math:`S/N` can carry :math:`C = B\log_2(1 + S/N)` bits per second
with arbitrarily small error (:func:`~engineeringkit.communications.shannon_capacity`),
and no more. Writing :math:`S/N = (E_b/N_0)(C/B)` gives the minimum energy
per bit for a given spectral efficiency. As bandwidth becomes unlimited the
minimum tends to :math:`\ln 2`, or -1.59 dB, the *Shannon limit*.

.. GENERATED FROM PYTHON SOURCE LINES 15-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.communications import db, shannon_capacity, spectral_efficiency








.. GENERATED FROM PYTHON SOURCE LINES 21-23

A telephone line and a Wi-Fi channel
------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 23-28

.. code-block:: Python


    for name, B, snr_db in [("telephone", 3.1e3, 35), ("Wi-Fi 20 MHz", 20e6, 25)]:
        C = float(shannon_capacity(B, 10 ** (snr_db / 10)))
        print(f"{name:14s}: C = {C / 1e3:9.1f} kbit/s, {float(spectral_efficiency(C, B)):.2f} bit/s/Hz")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    telephone     : C =      36.0 kbit/s, 11.63 bit/s/Hz
    Wi-Fi 20 MHz  : C =  166187.5 kbit/s, 8.31 bit/s/Hz




.. GENERATED FROM PYTHON SOURCE LINES 29-34

The minimum Eb/N0 for each spectral efficiency
----------------------------------------------

On the boundary :math:`\eta = \log_2(1 + \eta E_b/N_0)`, so
:math:`E_b/N_0 = (2^\eta - 1)/\eta`.

.. GENERATED FROM PYTHON SOURCE LINES 34-51

.. code-block:: Python


    eta = np.logspace(-3, np.log10(10), 200)
    ebn0_min = (2**eta - 1) / eta
    assert np.allclose(shannon_capacity(1.0, eta * ebn0_min), eta)
    print(f"Shannon limit as eta -> 0: {float(db(ebn0_min[0])):.3f} dB (ln 2 = {float(db(np.log(2))):.3f} dB)")
    fig, axes = plt.subplots(1, 2, figsize=(9, 3.6))
    snr = np.linspace(-10, 40, 200)
    axes[0].plot(snr, shannon_capacity(1.0, 10 ** (snr / 10)))
    axes[0].set(xlabel="S/N [dB]", ylabel="C / B [bit/s/Hz]", title="Capacity grows ~1 bit per 3 dB")
    axes[1].semilogy(db(ebn0_min), eta)
    axes[1].axvline(float(db(np.log(2))), color="gray", ls="--", label="-1.59 dB")
    axes[1].fill_betweenx(eta, db(ebn0_min), -2, color="C3", alpha=0.15, label="impossible")
    axes[1].set(xlim=(-2, 15), xlabel="$E_b/N_0$ [dB]", ylabel="spectral efficiency [bit/s/Hz]")
    axes[1].legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/communications/information/images/sphx_glr_plot_04_shannon_hartley_capacity_001.png
   :alt: Capacity grows ~1 bit per 3 dB
   :srcset: /api/gallery/communications/information/images/sphx_glr_plot_04_shannon_hartley_capacity_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Shannon limit as eta -> 0: -1.590 dB (ln 2 = -1.592 dB)





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.267 seconds)


.. _sphx_glr_download_api_gallery_communications_information_plot_04_shannon_hartley_capacity.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/communications/information/plot_04_shannon_hartley_capacity.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_04_shannon_hartley_capacity.ipynb <plot_04_shannon_hartley_capacity.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_04_shannon_hartley_capacity.py <plot_04_shannon_hartley_capacity.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_04_shannon_hartley_capacity.zip <plot_04_shannon_hartley_capacity.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_